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An open continuous injective map can also be described as a homeomorphism with an open subset. The stated property is obviously equivalent to: every open set in every open ball $B$ in $X$ contains an open set that is homeomorphic with $B$. If the metric is bounded, $B$ can be replaced by $X$ without loss of generality (and so the property becomes purely topological).

This property is a kind of local self-similarity (or just self-similarity in the bounded case).

Contrary to claims in the comments, the

The property does not hold for the one-point compactification of countably many disjoint closed intervals; contractible nor for contractible, locally contractible spaces, for instance it fails already for any finite tree with at least one vertex of degree $>2$ (also, the closed unit interval. It fails even "locally", locally" for the triod, in any reasonable the sense I can imagine)that the vertex of the triod has no neighborhood satisfying the property.

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An open continuous injective map can also be described as a homeomorphism with an open subset. The stated property is obviously equivalent to: every open set in every open ball $B$ in $X$ contains an open set that is homeomorphic with $B$. If the metric is bounded, $B$ can be replaced by $X$ without loss of generality (and so the property becomes purely topological).

This property is a kind of local self-similarity (or just self-similarity in the bounded case).

Contrary to claims in the comments, the property does not hold for the one-point compactification of countably many disjoint closed intervals; nor for contractible, locally contractible spaces, for instance it fails for any finite tree with at least one vertex of degree $>2$ (also, "locally", in any reasonable sense I can imagine).